**COMPARISON OF MEASURES OF CENTRAL TENDENCY**

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#### Central Tendency

**ABSTRACT**

This research is based on the comparisons of measures of central tendency. This is to find the most appropriate measures for average, using variance for easy collection of data of middle numbers that are centrally placed. Data was collected on which the various measures of central tendency was calculated. These measures were then used to calculate the variance for each distribution. This study shows that the arithmetic mean is the best measures of central tendency. It is therefore being recommended for wide usage as a reliable sample mean for population.

**TABLE OF CONTENT**

Chapter one: Introduction

Background to the study

Statement of problems

Objective of the study

Significance of study

Limitation of study

Definition of term

Chapter two: Literature review

Chapter three: Methodology

The mode

The median

The arithmetic mean

The geometric mean

The harmonic mean

The quadratic mean

Variance

Chapter four: Solved problem

Chapter five

Summary, conclusion and recommendation

Summary

Conclusion

Recommendation

References

**CHAPTER ONE**

BACKGROUND OF STUDY

In the study of statistics, descriptive statistics is one of the main area of statistics. It is concerned with the processing, summarizing, conclusions and presentation of qualitative and quantitative data. This area is limited to the data at hand no attempt is made to generalize when to begin courses in statistics with a discussion of so called descriptive statistics. That is, method of summarizing masses of numerical data, the graphical display of such data, procedures for ‘fitting’ data to theoretical model and so on. The aim of such procedures is that communicating, say to a colleague, the salient features of a set of numbers you have students. The assumption being that he does not want to be bothered with all facts which supports your conclusion, this is a laudable aim.

In addition, most rules o descriptive statistics depend on what one want to do with ones data, few people collect data in order to summarize, that they collect data to support arguments, or they collect data to draw inferences from them. If follow that the detailed of descriptive statistics should come after the study of statistical inference. After you know the kind of interference you can make from set of data, then you will know what calculations to performs, it reduces data to some form, usually a number that is easily comprehended.

A critical statistical notion for learners is that of data set as an entity, in other words, developing a statistical perspective. Holding statistical perspective requires a focus on the dataset as a collective rather than focusing on individual data values. By focusing on comparing of measure of central tendency. Would be provided with a conceptual structure that facilitates a focus on aggregates. Measures of central tendency are a descriptive statistics. If the value arranged in order of magnitude, a typical value will lie in the central region. This accounts for the reason why the measure of location is sometimes referred to as the measure of central tendency.

#### BACKGROUND TO THE STUDY

Measures of central tendency is also called measures of location or average. An average is a value that is typical or representatives of a set of data. Since such typical values tend to lie centrally within a set of data arranged according to magnitude. Practically everyone have used the word ‘average’ at one time or the other. By average we usually mean the ‘normal position., for example it is a common knowledge that ‘average’ age for primary school is between five and six years. Employers and workers union often quote what each other consider to be the ‘average’ income of employees but with that of the employer usually higher than that of the union suggesting that the two parties are often speaking of two different average though generated from the same body of data. Because of the ambiguity and the confusion associated with the term ‘average’.

#### STATEMENT OF PROBLEMS

Measures of central tendency are usually seen as the easiest part of descriptive statistics because it deals with day-to-day events. Lack of attention to distributional features of data is apparent in the dominance of numerical methods for making data comparison. Selecting and collection of data and finding of average numbers is not difficult but using terms like houses (real object) and living things seems difficult. The increased focus on elementary level data analysis and statistics is evident in the proliferation of statistical ideal in people.

#### OBJECTIVE OF THE STUDY

At the end of the research, we should be able to compare measures of central tendency in relation to variance. We would be able to know which among the measures of central tendency is best used in with the help of variance.

#### SIGNIFICANCE OF STUDY

At the end of the research, is application in the physical world would motivate students to study and apply it into real life situation.

This study is to also enhance the teaching and effective learning of this subject as it is taught in mathematics of an institution of higher learning. This is with special reference in the university, polytechnics and colleges of education. The study is also of importance to the statistician, scholars, the government and private sectors.

#### DEFINITION OF TERMS

Measures of central tendency of a distribution is a values which is typical or representation of the data. Since such typical values tend to lie centrally within a set of data arranged according to magnitude, average are also called measures of central tendency. The most common of measures of central tendency are the mode, the median and the arithmetic mean, the geometric mean, the harmonic mean and the quadratic mean.

The mode of a set of numbers is that value which has the highest frequency. It may not exist, and even if it does it may not be unique. A distribution having one mode is called unimodal.

The median of a distribution is that number for which the values of the distribution which are less than or equal to it is the same as the values which are greater than or equal to it. Is also a set of numbers arranged in order of magnitude (i.e. in an array, is either the middle value or the arithmetic mean of the two middle values.

The arithmetic mean of a set of numbers x_{1}, x_{2}, x_{3}, … x_{n} is their sum divided by n.

The geometric mean (G) of a set of N positive numbers x_{1}, x_{2}, x_{3}, … x_{n} is the nth root of the product of the numbers.

The harmonic mean (H) of a set of n number x_{1}, x_{2}, x_{3}, … x_{n }is the reciprocal of the arithmetic mean of the reciprocals of the numbers.

The quadratic mean of a set of square of number x_{1}, x_{2}, x_{3}, … x_{n} is their sum divided by n

The relations between the Arithmetic mean, Geometric mean, and Harmonic mean: The geometric mean of a set of positive number x_{1}, x_{2}, x_{3}, … x_{n} is less than or equal to their arithmetic mean but is greater than or equal to their harmonic mean.

Variance of a set of numbers is the arithmetic mean of the squares of deviation from the mean of numbers.